So is it better to say 'He asked the girl not to bother Mr. Craven' ?
Can I split the infinitive or is that gramatically incorrect?
And does this happen always when you have ask for and a negative sentence (to use not to)?

Ah, thanks for the correction. I guess it's just one of those finer points of grammar that is usually never contemplated by native speakers, as i use and hear both all the time.

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Don't worry. It's one of those rules that do not really reflect linguistic reality anymore. You are right: you hear/see both constructions. It's only when you have to do an exam and you happen to have a traditional corrector that you may get in trouble.